2,484 research outputs found
The valuation criterion for normal basis generators
If is a finite Galois extension of local fields, we say that the
valuation criterion holds if there is an integer such that every
element with valuation generates a normal basis for .
Answering a question of Byott and Elder, we first prove that holds if
and only if the tamely ramified part of the extension is trivial and
every non-zero -submodule of contains a unit. Moreover, the integer
can take one value modulo only, namely , where
is the valuation of the different of . When has positive
characteristic, we thus recover a recent result of Elder and Thomas, proving
that is valid for all extensions in this context. When
\char{\;K}=0, we identify all abelian extensions for which is
true, using algebraic arguments. These extensions are determined by the
behaviour of their cyclic Kummer subextensions
Pointed Hopf actions on fields, I
Actions of semisimple Hopf algebras H over an algebraically closed field of
characteristic zero on commutative domains were classified recently by the
authors. The answer turns out to be very simple- if the action is inner
faithful, then H has to be a group algebra. The present article contributes to
the non-semisimple case, which is much more complicated. Namely, we study
actions of finite dimensional (not necessarily semisimple) Hopf algebras on
commutative domains, particularly when H is pointed of finite Cartan type.
The work begins by reducing to the case where H acts inner faithfully on a
field; such a Hopf algebra is referred to as Galois-theoretical. We present
examples of such Hopf algebras, which include the Taft algebras, u_q(sl_2), and
some Drinfeld twists of other small quantum groups. We also give many examples
of finite dimensional Hopf algebras which are not Galois-theoretical.
Classification results on finite dimensional pointed Galois-theoretical Hopf
algebras of finite Cartan type will be provided in the sequel, Part II, of this
study.Comment: v4: This version is unchanged from v3. This article has appeared in
Transformation Groups. The TG reference numbers versus the arxiv reference
numbers are available in the appendix (Section 5) of this versio
The fundamental group of a Hopf linear category
We define the fundamental group of a Hopf algebra over a field. For this
purpose we first consider gradings of Hopf algebras and Galois coverings. The
latter are given by linear categories with new additional structure which we
call Hopf linear categories over a finite group. We compare this invariant to
the fundamental group of the underlying linear category, and we compute those
groups for families of examples.Comment: Computations of the fundamental group of some Hopf algebras are
added. The relation with the fundamental group of the underlying associative
structure is now considered. We also analyse the situation when universal
covers and/or gradings exist. Dedicated to Eduardo N. Marcos for his 60th
birthday. 24 page
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